# Properties

 Label 560.2.q.h Level $560$ Weight $2$ Character orbit 560.q Analytic conductor $4.472$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$560 = 2^{4} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 560.q (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.47162251319$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 280) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{6} + 1) q^{3} + \zeta_{6} q^{5} + ( - \zeta_{6} + 3) q^{7} + 2 \zeta_{6} q^{9}+O(q^{10})$$ q + (-z + 1) * q^3 + z * q^5 + (-z + 3) * q^7 + 2*z * q^9 $$q + ( - \zeta_{6} + 1) q^{3} + \zeta_{6} q^{5} + ( - \zeta_{6} + 3) q^{7} + 2 \zeta_{6} q^{9} + ( - 2 \zeta_{6} + 2) q^{11} + q^{15} + (4 \zeta_{6} - 4) q^{17} - 2 \zeta_{6} q^{19} + ( - 3 \zeta_{6} + 2) q^{21} + \zeta_{6} q^{23} + (\zeta_{6} - 1) q^{25} + 5 q^{27} + 9 q^{29} + ( - 4 \zeta_{6} + 4) q^{31} - 2 \zeta_{6} q^{33} + (2 \zeta_{6} + 1) q^{35} - 4 \zeta_{6} q^{37} + q^{41} - 9 q^{43} + (2 \zeta_{6} - 2) q^{45} + ( - 5 \zeta_{6} + 8) q^{49} + 4 \zeta_{6} q^{51} + ( - 10 \zeta_{6} + 10) q^{53} + 2 q^{55} - 2 q^{57} + (10 \zeta_{6} - 10) q^{59} - 9 \zeta_{6} q^{61} + (4 \zeta_{6} + 2) q^{63} + ( - 5 \zeta_{6} + 5) q^{67} + q^{69} - 14 q^{71} + (12 \zeta_{6} - 12) q^{73} + \zeta_{6} q^{75} + ( - 6 \zeta_{6} + 4) q^{77} + 14 \zeta_{6} q^{79} + (\zeta_{6} - 1) q^{81} - 11 q^{83} - 4 q^{85} + ( - 9 \zeta_{6} + 9) q^{87} + 15 \zeta_{6} q^{89} - 4 \zeta_{6} q^{93} + ( - 2 \zeta_{6} + 2) q^{95} - 18 q^{97} + 4 q^{99} +O(q^{100})$$ q + (-z + 1) * q^3 + z * q^5 + (-z + 3) * q^7 + 2*z * q^9 + (-2*z + 2) * q^11 + q^15 + (4*z - 4) * q^17 - 2*z * q^19 + (-3*z + 2) * q^21 + z * q^23 + (z - 1) * q^25 + 5 * q^27 + 9 * q^29 + (-4*z + 4) * q^31 - 2*z * q^33 + (2*z + 1) * q^35 - 4*z * q^37 + q^41 - 9 * q^43 + (2*z - 2) * q^45 + (-5*z + 8) * q^49 + 4*z * q^51 + (-10*z + 10) * q^53 + 2 * q^55 - 2 * q^57 + (10*z - 10) * q^59 - 9*z * q^61 + (4*z + 2) * q^63 + (-5*z + 5) * q^67 + q^69 - 14 * q^71 + (12*z - 12) * q^73 + z * q^75 + (-6*z + 4) * q^77 + 14*z * q^79 + (z - 1) * q^81 - 11 * q^83 - 4 * q^85 + (-9*z + 9) * q^87 + 15*z * q^89 - 4*z * q^93 + (-2*z + 2) * q^95 - 18 * q^97 + 4 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} + q^{5} + 5 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q + q^3 + q^5 + 5 * q^7 + 2 * q^9 $$2 q + q^{3} + q^{5} + 5 q^{7} + 2 q^{9} + 2 q^{11} + 2 q^{15} - 4 q^{17} - 2 q^{19} + q^{21} + q^{23} - q^{25} + 10 q^{27} + 18 q^{29} + 4 q^{31} - 2 q^{33} + 4 q^{35} - 4 q^{37} + 2 q^{41} - 18 q^{43} - 2 q^{45} + 11 q^{49} + 4 q^{51} + 10 q^{53} + 4 q^{55} - 4 q^{57} - 10 q^{59} - 9 q^{61} + 8 q^{63} + 5 q^{67} + 2 q^{69} - 28 q^{71} - 12 q^{73} + q^{75} + 2 q^{77} + 14 q^{79} - q^{81} - 22 q^{83} - 8 q^{85} + 9 q^{87} + 15 q^{89} - 4 q^{93} + 2 q^{95} - 36 q^{97} + 8 q^{99}+O(q^{100})$$ 2 * q + q^3 + q^5 + 5 * q^7 + 2 * q^9 + 2 * q^11 + 2 * q^15 - 4 * q^17 - 2 * q^19 + q^21 + q^23 - q^25 + 10 * q^27 + 18 * q^29 + 4 * q^31 - 2 * q^33 + 4 * q^35 - 4 * q^37 + 2 * q^41 - 18 * q^43 - 2 * q^45 + 11 * q^49 + 4 * q^51 + 10 * q^53 + 4 * q^55 - 4 * q^57 - 10 * q^59 - 9 * q^61 + 8 * q^63 + 5 * q^67 + 2 * q^69 - 28 * q^71 - 12 * q^73 + q^75 + 2 * q^77 + 14 * q^79 - q^81 - 22 * q^83 - 8 * q^85 + 9 * q^87 + 15 * q^89 - 4 * q^93 + 2 * q^95 - 36 * q^97 + 8 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/560\mathbb{Z}\right)^\times$$.

 $$n$$ $$241$$ $$337$$ $$351$$ $$421$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$ $$1$$ $$1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
81.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0.500000 0.866025i 0 0.500000 + 0.866025i 0 2.50000 0.866025i 0 1.00000 + 1.73205i 0
401.1 0 0.500000 + 0.866025i 0 0.500000 0.866025i 0 2.50000 + 0.866025i 0 1.00000 1.73205i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 560.2.q.h 2
4.b odd 2 1 280.2.q.a 2
7.c even 3 1 inner 560.2.q.h 2
7.c even 3 1 3920.2.a.m 1
7.d odd 6 1 3920.2.a.y 1
12.b even 2 1 2520.2.bi.a 2
20.d odd 2 1 1400.2.q.e 2
20.e even 4 2 1400.2.bh.b 4
28.d even 2 1 1960.2.q.k 2
28.f even 6 1 1960.2.a.e 1
28.f even 6 1 1960.2.q.k 2
28.g odd 6 1 280.2.q.a 2
28.g odd 6 1 1960.2.a.i 1
84.n even 6 1 2520.2.bi.a 2
140.p odd 6 1 1400.2.q.e 2
140.p odd 6 1 9800.2.a.r 1
140.s even 6 1 9800.2.a.bc 1
140.w even 12 2 1400.2.bh.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.q.a 2 4.b odd 2 1
280.2.q.a 2 28.g odd 6 1
560.2.q.h 2 1.a even 1 1 trivial
560.2.q.h 2 7.c even 3 1 inner
1400.2.q.e 2 20.d odd 2 1
1400.2.q.e 2 140.p odd 6 1
1400.2.bh.b 4 20.e even 4 2
1400.2.bh.b 4 140.w even 12 2
1960.2.a.e 1 28.f even 6 1
1960.2.a.i 1 28.g odd 6 1
1960.2.q.k 2 28.d even 2 1
1960.2.q.k 2 28.f even 6 1
2520.2.bi.a 2 12.b even 2 1
2520.2.bi.a 2 84.n even 6 1
3920.2.a.m 1 7.c even 3 1
3920.2.a.y 1 7.d odd 6 1
9800.2.a.r 1 140.p odd 6 1
9800.2.a.bc 1 140.s even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(560, [\chi])$$:

 $$T_{3}^{2} - T_{3} + 1$$ T3^2 - T3 + 1 $$T_{11}^{2} - 2T_{11} + 4$$ T11^2 - 2*T11 + 4 $$T_{13}$$ T13

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - T + 1$$
$5$ $$T^{2} - T + 1$$
$7$ $$T^{2} - 5T + 7$$
$11$ $$T^{2} - 2T + 4$$
$13$ $$T^{2}$$
$17$ $$T^{2} + 4T + 16$$
$19$ $$T^{2} + 2T + 4$$
$23$ $$T^{2} - T + 1$$
$29$ $$(T - 9)^{2}$$
$31$ $$T^{2} - 4T + 16$$
$37$ $$T^{2} + 4T + 16$$
$41$ $$(T - 1)^{2}$$
$43$ $$(T + 9)^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2} - 10T + 100$$
$59$ $$T^{2} + 10T + 100$$
$61$ $$T^{2} + 9T + 81$$
$67$ $$T^{2} - 5T + 25$$
$71$ $$(T + 14)^{2}$$
$73$ $$T^{2} + 12T + 144$$
$79$ $$T^{2} - 14T + 196$$
$83$ $$(T + 11)^{2}$$
$89$ $$T^{2} - 15T + 225$$
$97$ $$(T + 18)^{2}$$